304 35. STABILITY OF RICCI FLOW
solutions relevant to geometrization), stationary solutions of this type are stable at-
tractors: solutions originating from initial data in a 1 +a little Holder neighborhood
of a stationary solution converge to that solution exponentially fast in a 2+/3 Holder
norm [162].
4. Dynamic stability results obtained by other methods
4.1. Dynamic stability in the presence of positive curvature.
In this subsection, we briefly review various convergence theorems for Ricci flow
obtained by methods other than linearization. For those cases in which a unique
limit exists, each convergence theorem implies a stability theorem- though all a re
strictly stronger since none assumes existence of a stationary solution.
REMARK 35.34. We have inserted the hypoth esis of simple connectivity in
Theorems 35.35- 35.37 and 35.39- 35.40 below to ensure uniqueness (up to isometry)
of the resulting limits.
Hamilton's seminal result for Ricci fl.ow in dimension n = 3 implies a stability
theorem because {g : Rc(g) > 0} is an open neighborhood of a metric of constant
positive sectional curvature.THEOREM 35.35 ([135]). If (M^3 ,g) is a closed simply-connected Riemannian
manifold of positive Ricci curvature, th en the volume normalized Ricci flow with
initial data g converges to a unique metric of constant positive curvature.Of course, Ha milton's theorem yields a much stronger result since one does not
know a priori that M^3 admits a co nstant sectional curvature metric, i.e., that M^3
is a space form.
Hamilton also proves the following in dimension n = 4.
THEOREM 35.36 ([136]). If (M^4 ,g) is a closed simply-connected Riemannian
manifold with positive curvature operator, then the volume normalized Ricci flow
with initial data g converges to a unique metric of constant positive curvature. In
particular, M^4 is diffeomorphic to the standard 4-sphere if it is orientable and to
!RlP'^4 otherwise.
Instead of assuming positivity of a curvature quantity, one may also impose
a pinching condition. Huisken's pinching theorem for dimensions n ;:=:: 4 implies
stability of metrics of co nstant positive sectional curvature in those dimensions. To
state it, we recall the orthogonal decomposition of the Riemann curvature tensor
Rm=U+V+W
into irreducible components
(35.28) U = 2n (: _ 1) (g !\ g) '
1 0
V =--(Re!\ g),
n-2W = Weyl tensor,
0
where Re denotes the trace-free Ricci tensor and A denotes the Kulkarni- Nomizu
product of symmetric tensors:
(h !\ k)(X1, X2, X3, X 4) ~ h(X1, X4)k(X2, X3) + h(X2, X 3)k(X1, X4)
- h(X1, X 3)k(X2, X4) - h(X2, X4)k(X1, X3).